The purpose of this study is to assess the safety of combination treatment of GSK2256098 and trametinib in mesothelioma subjects and subjects with other selected tumor types. Also, the study will identify a maximum tolerated combination dose of GSK2256098 and trametinib. This study is a Phase I, open-label, dose-escalation study to determine maximal tolerated dose (MTD) and the recommended Phase 2 dose (RP2D) and regimens for oral MEK inhibitor trametinib (once daily [OD]dosing) and the oral FAK inhibitor GSK2256098 (twice daily [BID] dosing). The synergy of the combination was observed over a wide range of concentrations and results in several-fold reduction in compound concentration to achieve equivalent biological responses compared to either single agent. The dose and schedule of dosing may be modified based on emerging safety, pharmacokinetic (PK), and pharmacodynamic (PD) data. The study will be conducted in two parts; Part 1 Dose Escalation to determine the MTD and RP2D and Part 2 Expansion Cohort to further evaluate the safety and tolerability of trametinib and GSK2256098 at the RP2D and determine clinical activity. Additionally, in Part 1 Dose Escalation, additional subjects with malignant pleural mesothelioma (MPM) will be recruited at doses that are considered tolerable in order to assess PD in MPM subjects at each dose (the Pharmacodynamic Cohort). The Expansion Cohort will be limited to subjects with MPM who have progressed or are intolerant to first-line therapy.